In addition to rotary tables used to rotate drilling pipe in well drilling operations, top drive motor units are also used to rotate drilling pipe. As sections of pipe are tripped into or out of the drill string on drilling rigs having top drive units, it is desirable to have a pipe clamp sub-assembly on the top drive unit that is capable of gripping a section of pipe and drawing it towards or away from the quill of the top drive unit when making or breaking connections between the pipe and the top drive unit.